Chapter 3
Using Derivatives
3.1 Using derivatives to identify extreme values
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What are the critical numbers of a function f and how are they connected to
identifying the most extreme values the function achieves?
• How does the first derivative of a function reveal important information about the
behavior of the function, including the function’s extreme values?
• How can the second derivative of a function be used to help identify extreme
values of the function?
Introduction
In many different settings, we are interested in knowing where a function achieves its
least and greatest values. These can be important in applications – say to identify a point
at which maximum profit or minimum cost occurs – or in theory to understand how to
characterize the behavior of a function or a family of related functions. Consider the
simple and familiar example of a parabolic function such as s(t) = −16t 2 + 32t + 48 (shown
at left in Figure 3.1) that represents the height of an object tossed vertically: its maximum
value occurs at the vertex of the parabola and represents the highest value that the object
reaches. Moreover, this maximum value identifies an especially important point on the
graph, the point at which the curve changes from increasing to decreasing.
More generally, for any function we consider, we can investigate where its lowest
161
Using Derivatives
3.1 Using derivatives to identify extreme values
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What are the critical numbers of a function f and how are they connected to
identifying the most extreme values the function achieves?
• How does the first derivative of a function reveal important information about the
behavior of the function, including the function’s extreme values?
• How can the second derivative of a function be used to help identify extreme
values of the function?
Introduction
In many different settings, we are interested in knowing where a function achieves its
least and greatest values. These can be important in applications – say to identify a point
at which maximum profit or minimum cost occurs – or in theory to understand how to
characterize the behavior of a function or a family of related functions. Consider the
simple and familiar example of a parabolic function such as s(t) = −16t 2 + 32t + 48 (shown
at left in Figure 3.1) that represents the height of an object tossed vertically: its maximum
value occurs at the vertex of the parabola and represents the highest value that the object
reaches. Moreover, this maximum value identifies an especially important point on the
graph, the point at which the curve changes from increasing to decreasing.
More generally, for any function we consider, we can investigate where its lowest
161